The position vectors of P, Q and R are 2a – b, m(a – b), and a + b respectively. (a) Find the value of m if PQR is a straight line.
(b) State the ratio PQ:QR.
To determine the values of m and the ratio PQ:QR, we need to use the fact that PQR is a straight line. In a straight line, the position vectors of three points are collinear, meaning that they lie on the same line. Therefore, the direction vector of any two position vectors on the line should be proportional.
(a) To find the value of m, we need to find the direction vectors of PQ and QR and check if they are proportional.
The direction vector of a line between two points is obtained by subtracting the initial position vector from the final position vector.
Direction vector of PQ = (m(a - b)) - (2a - b) = -2a + (m - 1)b
Direction vector of QR = (a + b) - (m(a - b)) = (1 - ma) + (m + 1)b
For these two direction vectors to be proportional, their components must be multiples of each other.
Therefore, we can equate the x, y, and z components to create a system of equations:
-2a = (1 - ma)
m - 1 = m + 1
b = b
Simplifying these equations, we get:
-2a = 1 - ma --> [1]
-1 = 2 --> [2]
b = b --> [3]
From equation [2], we see that it is not possible for -1 to equal 2. Therefore, we can conclude that there is no value of m that makes PQR a straight line.
(b) Since there is no straight line, it is not possible to find the ratio PQ:QR.